## Abstract

Let $F$ be a strictly $k$-balanced $k$-uniform hypergraph with $e(F)\geq |F|-k+1$ and maximum co-degree at least two.

The random greedy $F$-free process constructs a maximal $F$-free hypergraph as follows.

Consider a random ordering of the hyperedges of the complete $k$-uniform hypergraph $K_n^k$ on $n$ vertices.

Start with the empty hypergraph on $n$ vertices. Successively consider the hyperedges $e$ of $K_n^k$ in the given ordering and add $e$ to the

existing hypergraph provided that $e$ does not create a copy of $F$.

We show that asymptotically almost surely this process terminates at a hypergraph with $\tilde{O}(n^{k-(|F|-k)/(e(F)-1)})$ hyperedges. This is best possible up to logarithmic factors.

The random greedy $F$-free process constructs a maximal $F$-free hypergraph as follows.

Consider a random ordering of the hyperedges of the complete $k$-uniform hypergraph $K_n^k$ on $n$ vertices.

Start with the empty hypergraph on $n$ vertices. Successively consider the hyperedges $e$ of $K_n^k$ in the given ordering and add $e$ to the

existing hypergraph provided that $e$ does not create a copy of $F$.

We show that asymptotically almost surely this process terminates at a hypergraph with $\tilde{O}(n^{k-(|F|-k)/(e(F)-1)})$ hyperedges. This is best possible up to logarithmic factors.

Original language | English |
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Pages (from-to) | 1343-1350 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 30 |

Issue number | 3 |

Early online date | 12 Jul 2016 |

DOIs | |

Publication status | Published - 2016 |